Abstract
In this paper, we study the prize-collecting k-Steiner tree problem (PC k-ST), which is an interesting generalization of both the k-Steiner tree problem (k-ST) and the prize-collecting Steiner tree problem (PCST). In the PC k-ST, we are given an undirected connected graph \(G =(V, E)\), a subset \(R \subseteq V\) called terminals, a root vertex \(r \in V\) and an integer k. Every edge has a non-negative edge cost and every vertex has a non-negative penalty cost. We wish to find an r-rooted tree F that spans at least k vertices in R so as to minimize the total edge costs of F as well as the penalty costs of the vertices not in F. As our main contribution, we propose two approximation algorithms for the PC k-ST with ratios of 5.9672 and 5. The first algorithm is based on an observation of the solutions for the k-ST and the PCST, and the second one is based on the technique of primal-dual.
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Acknowledgements
The first author is supported by the National Natural Science Foundation of China (No. 12001523). The second author is supported by Scientific Research Project of Beijing Municipal Education Commission (No. KM201910005012) and the National Natural Science Foundation of China (No. 11971046). The third and fourth authors are supported by the National Natural Science Foundation of China (No. 11871081). The third author is also supported by Beijing Natural Science Foundation Project No. Z200002.
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Han, L., Wang, C., Xu, D., Zhang, D. (2021). The Prize-Collecting k-Steiner Tree Problem. In: Zhang, Y., Xu, Y., Tian, H. (eds) Parallel and Distributed Computing, Applications and Technologies. PDCAT 2020. Lecture Notes in Computer Science(), vol 12606. Springer, Cham. https://doi.org/10.1007/978-3-030-69244-5_33
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